A300813 a(n) is the smallest multiple of A001414(a(n-1)) not yet seen in the sequence; a(1)=2.

2, 4, 8, 6, 5, 10, 7, 14, 9, 12, 21, 20, 18, 16, 24, 27, 36, 30, 40, 11, 22, 13, 26, 15, 32, 50, 48, 33, 28, 44, 45, 55, 64, 60, 72, 84, 42, 96, 39, 80, 52, 17, 34, 19, 38, 63, 65, 54, 66, 112, 75, 78, 90, 91, 100, 56, 104, 57, 88, 51, 120, 70, 98, 128, 126, 105, 135, 140, 144, 154, 160, 150, 165, 76, 23

Offset: 1

Author

David James Sycamore, Mar 13 2018

Keywords

• nonn
• look

Comments

If for some n, A001414(a(n-1)) is a prime p not seen before, then a(n)=p and a(n+1)=2*p. If the lesser of a twin prime pair is a(n), the greater is a(n+2).

The numbers not appearing in the first 10^7 terms are 1, 3, 37957, 37963, 38557, 39301, 40237, 40343, 40351, 40357, ...; it seems that all numbers other than 1 and 3 eventually appear. - Charles R Greathouse IV, Apr 09 2018

Examples

			a(2)=4 because A001414(2)=2 and 4 is the least multiple of 2 not yet seen.
a(3)=8 because A001414(4)=4 and 8 is the least multiple of 4 not seen yet.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Charles R Greathouse IV, Table of n, a(n) for n = 1..702396

Crossrefs

Cf. A001414.

Programs

Mathematica

Nest[Append[#, Block[{k = 1, m = Total@ Flatten@ Map[ConstantArray[#1, #2] & @@ # &, FactorInteger@ Last@ #]}, While[! FreeQ[#, k m], k++]; k m]] &, {2}, 74] (* Michael De Vlieger, Mar 14 2018 *)

PARI

sopfr(n,f=factor(n))=sum(i=1,#f[,1],f[i,1]*f[i,2])
first(n)=my(v=vector(n),s=[2],t); v[1]=2; for(m=2,n, t=sopfr(v[m-1]); forstep(k=t,t*m,t, if(setsearch(s,k), next); s=setunion(s,[k]); v[m]=k; break)); v \\ Charles R Greathouse IV, Apr 06 2018